------------------------------------------------------------------------
-- The Agda standard library
--
-- Consequences of a monomorphism between ring-like structures
------------------------------------------------------------------------

-- See Data.Nat.Binary.Properties for examples of how this and similar
-- modules can be used to easily translate properties between types.

{-# OPTIONS --cubical-compatible --safe #-}

open import Algebra.Bundles using (RawRing)
open import Algebra.Morphism.Structures using (IsRingMonomorphism)


module Algebra.Morphism.RingMonomorphism
  {a b ℓ₁ ℓ₂} {R₁ : RawRing a ℓ₁} {R₂ : RawRing b ℓ₂} {⟦_⟧}
  (isRingMonomorphism : IsRingMonomorphism R₁ R₂ ⟦_⟧)
  where

open IsRingMonomorphism isRingMonomorphism
open RawRing R₁ renaming (Carrier to A; _≈_ to _≈₁_)
open RawRing R₂ renaming
  ( Carrier to B; _≈_ to _≈₂_; _+_ to _⊕_
  ; _*_ to _⊛_; 1# to 1#₂; 0# to 0#₂; -_ to ⊝_)
import Algebra.Morphism.GroupMonomorphism  as GroupMonomorphism
import Algebra.Morphism.MonoidMonomorphism as MonoidMonomorphism
open import Relation.Binary.Core using (Rel)
open import Algebra.Definitions
  using (_DistributesOverˡ_; _DistributesOverʳ_ ; _DistributesOver_
        ; LeftZero; RightZero; Zero)
open import Algebra.Structures
  using (IsRing; IsCommutativeRing; IsGroup; IsMagma)
open import Data.Product.Base using (proj₁; proj₂; _,_)
import Relation.Binary.Reasoning.Setoid as ≈-Reasoning

------------------------------------------------------------------------
-- Re-export all properties of group and monoid monomorphisms

open GroupMonomorphism +-isGroupMonomorphism public
  renaming
  ( assoc   to +-assoc
  ; comm    to +-comm
  ; cong    to +-cong
  ; idem    to +-idem
  ; sel     to +-sel
  ; ⁻¹-cong to neg-cong

  ; identity to +-identity; identityˡ to +-identityˡ; identityʳ to +-identityʳ
  ; cancel   to +-cancel;   cancelˡ   to +-cancelˡ;   cancelʳ   to +-cancelʳ
  ; zero     to +-zero;     zeroˡ     to +-zeroˡ;     zeroʳ     to +-zeroʳ

  ; isMagma             to +-isMagma
  ; isSemigroup         to +-isSemigroup
  ; isMonoid            to +-isMonoid
  ; isSelectiveMagma    to +-isSelectiveMagma
  ; isBand              to +-isBand
  ; isCommutativeMonoid to +-isCommutativeMonoid
  )

open MonoidMonomorphism *-isMonoidMonomorphism public
  renaming
  ( assoc to *-assoc
  ; comm  to *-comm
  ; cong  to *-cong
  ; idem  to *-idem
  ; sel   to *-sel

  ; identity to *-identity; identityˡ to *-identityˡ; identityʳ to *-identityʳ
  ; cancel   to *-cancel;   cancelˡ   to *-cancelˡ;   cancelʳ   to *-cancelʳ
  ; zero     to *-zero;     zeroˡ     to *-zeroˡ;     zeroʳ     to *-zeroʳ

  ; isMagma             to *-isMagma
  ; isSemigroup         to *-isSemigroup
  ; isMonoid            to *-isMonoid
  ; isSelectiveMagma    to *-isSelectiveMagma
  ; isBand              to *-isBand
  ; isCommutativeMonoid to *-isCommutativeMonoid
  )

------------------------------------------------------------------------
-- Properties

module _ (+-isGroup : IsGroup _≈₂_ _⊕_ 0#₂ ⊝_)
         (*-isMagma : IsMagma _≈₂_ _⊛_) where

  open IsGroup +-isGroup hiding (setoid; refl; sym)
  open IsMagma *-isMagma renaming (∙-cong to ◦-cong)
  open ≈-Reasoning setoid

  distribˡ : _DistributesOverˡ_ _≈₂_ _⊛_ _⊕_ → _DistributesOverˡ_ _≈₁_ _*_ _+_
  distribˡ distribˡ x y z = injective (begin
    $\begin{pmatrix} \,\href{Algebra.Morphism.RingMonomorphism.html#3306}{\htmlId{3337}{\htmlClass{Bound}{\text{x}}}}\, \,\href{Algebra.Bundles.Raw.html#5431}{\htmlId{3339}{\htmlClass{Function Operator}{\text{*}}}}\, \,\htmlId{3341}{\htmlClass{Symbol}{\text{(}}}\,\,\href{Algebra.Morphism.RingMonomorphism.html#3308}{\htmlId{3342}{\htmlClass{Bound}{\text{y}}}}\, \,\href{Algebra.Bundles.Raw.html#5405}{\htmlId{3344}{\htmlClass{Function Operator}{\text{+}}}}\, \,\href{Algebra.Morphism.RingMonomorphism.html#3310}{\htmlId{3346}{\htmlClass{Bound}{\text{z}}}}\,\,\htmlId{3347}{\htmlClass{Symbol}{\text{)}}}\, \end{pmatrix}$               ≈⟨ *-homo x (y + z) ⟩
    $\begin{pmatrix} \,\href{Algebra.Morphism.RingMonomorphism.html#3306}{\htmlId{3393}{\htmlClass{Bound}{\text{x}}}}\, \end{pmatrix}$ ⊛ $\begin{pmatrix} \,\href{Algebra.Morphism.RingMonomorphism.html#3308}{\htmlId{3401}{\htmlClass{Bound}{\text{y}}}}\, \,\href{Algebra.Bundles.Raw.html#5405}{\htmlId{3403}{\htmlClass{Function Operator}{\text{+}}}}\, \,\href{Algebra.Morphism.RingMonomorphism.html#3310}{\htmlId{3405}{\htmlClass{Bound}{\text{z}}}}\, \end{pmatrix}$             ≈⟨ ◦-cong refl (+-homo y z) ⟩
    $\begin{pmatrix} \,\href{Algebra.Morphism.RingMonomorphism.html#3306}{\htmlId{3457}{\htmlClass{Bound}{\text{x}}}}\, \end{pmatrix}$ ⊛ ($\begin{pmatrix} \,\href{Algebra.Morphism.RingMonomorphism.html#3308}{\htmlId{3466}{\htmlClass{Bound}{\text{y}}}}\, \end{pmatrix}$ ⊕ $\begin{pmatrix} \,\href{Algebra.Morphism.RingMonomorphism.html#3310}{\htmlId{3474}{\htmlClass{Bound}{\text{z}}}}\, \end{pmatrix}$)       ≈⟨ distribˡ $\begin{pmatrix} \,\href{Algebra.Morphism.RingMonomorphism.html#3306}{\htmlId{3499}{\htmlClass{Bound}{\text{x}}}}\, \end{pmatrix}$ $\begin{pmatrix} \,\href{Algebra.Morphism.RingMonomorphism.html#3308}{\htmlId{3505}{\htmlClass{Bound}{\text{y}}}}\, \end{pmatrix}$ $\begin{pmatrix} \,\href{Algebra.Morphism.RingMonomorphism.html#3310}{\htmlId{3511}{\htmlClass{Bound}{\text{z}}}}\, \end{pmatrix}$ ⟩
    $\begin{pmatrix} \,\href{Algebra.Morphism.RingMonomorphism.html#3306}{\htmlId{3523}{\htmlClass{Bound}{\text{x}}}}\, \end{pmatrix}$ ⊛ $\begin{pmatrix} \,\href{Algebra.Morphism.RingMonomorphism.html#3308}{\htmlId{3531}{\htmlClass{Bound}{\text{y}}}}\, \end{pmatrix}$ ⊕ $\begin{pmatrix} \,\href{Algebra.Morphism.RingMonomorphism.html#3306}{\htmlId{3539}{\htmlClass{Bound}{\text{x}}}}\, \end{pmatrix}$ ⊛ $\begin{pmatrix} \,\href{Algebra.Morphism.RingMonomorphism.html#3310}{\htmlId{3547}{\htmlClass{Bound}{\text{z}}}}\, \end{pmatrix}$ ≈⟨ ∙-cong (*-homo x y) (*-homo x z) ⟨
    $\begin{pmatrix} \,\href{Algebra.Morphism.RingMonomorphism.html#3306}{\htmlId{3595}{\htmlClass{Bound}{\text{x}}}}\, \,\href{Algebra.Bundles.Raw.html#5431}{\htmlId{3597}{\htmlClass{Function Operator}{\text{*}}}}\, \,\href{Algebra.Morphism.RingMonomorphism.html#3308}{\htmlId{3599}{\htmlClass{Bound}{\text{y}}}}\, \end{pmatrix}$ ⊕ $\begin{pmatrix} \,\href{Algebra.Morphism.RingMonomorphism.html#3306}{\htmlId{3607}{\htmlClass{Bound}{\text{x}}}}\, \,\href{Algebra.Bundles.Raw.html#5431}{\htmlId{3609}{\htmlClass{Function Operator}{\text{*}}}}\, \,\href{Algebra.Morphism.RingMonomorphism.html#3310}{\htmlId{3611}{\htmlClass{Bound}{\text{z}}}}\, \end{pmatrix}$         ≈⟨ +-homo (x * y) (x * z) ⟨
    $\begin{pmatrix} \,\href{Algebra.Morphism.RingMonomorphism.html#3306}{\htmlId{3657}{\htmlClass{Bound}{\text{x}}}}\, \,\href{Algebra.Bundles.Raw.html#5431}{\htmlId{3659}{\htmlClass{Function Operator}{\text{*}}}}\, \,\href{Algebra.Morphism.RingMonomorphism.html#3308}{\htmlId{3661}{\htmlClass{Bound}{\text{y}}}}\, \,\href{Algebra.Bundles.Raw.html#5405}{\htmlId{3663}{\htmlClass{Function Operator}{\text{+}}}}\, \,\href{Algebra.Morphism.RingMonomorphism.html#3306}{\htmlId{3665}{\htmlClass{Bound}{\text{x}}}}\, \,\href{Algebra.Bundles.Raw.html#5431}{\htmlId{3667}{\htmlClass{Function Operator}{\text{*}}}}\, \,\href{Algebra.Morphism.RingMonomorphism.html#3310}{\htmlId{3669}{\htmlClass{Bound}{\text{z}}}}\, \end{pmatrix}$ ∎)

  distribʳ : _DistributesOverʳ_ _≈₂_ _⊛_ _⊕_ → _DistributesOverʳ_ _≈₁_ _*_ _+_
  distribʳ distribˡ x y z = injective (begin
    $\begin{pmatrix} \,\htmlId{3807}{\htmlClass{Symbol}{\text{(}}}\,\,\href{Algebra.Morphism.RingMonomorphism.html#3778}{\htmlId{3808}{\htmlClass{Bound}{\text{y}}}}\, \,\href{Algebra.Bundles.Raw.html#5405}{\htmlId{3810}{\htmlClass{Function Operator}{\text{+}}}}\, \,\href{Algebra.Morphism.RingMonomorphism.html#3780}{\htmlId{3812}{\htmlClass{Bound}{\text{z}}}}\,\,\htmlId{3813}{\htmlClass{Symbol}{\text{)}}}\, \,\href{Algebra.Bundles.Raw.html#5431}{\htmlId{3815}{\htmlClass{Function Operator}{\text{*}}}}\, \,\href{Algebra.Morphism.RingMonomorphism.html#3776}{\htmlId{3817}{\htmlClass{Bound}{\text{x}}}}\, \end{pmatrix}$               ≈⟨ *-homo (y + z) x ⟩
    $\begin{pmatrix} \,\href{Algebra.Morphism.RingMonomorphism.html#3778}{\htmlId{3863}{\htmlClass{Bound}{\text{y}}}}\, \,\href{Algebra.Bundles.Raw.html#5405}{\htmlId{3865}{\htmlClass{Function Operator}{\text{+}}}}\, \,\href{Algebra.Morphism.RingMonomorphism.html#3780}{\htmlId{3867}{\htmlClass{Bound}{\text{z}}}}\, \end{pmatrix}$ ⊛ $\begin{pmatrix} \,\href{Algebra.Morphism.RingMonomorphism.html#3776}{\htmlId{3875}{\htmlClass{Bound}{\text{x}}}}\, \end{pmatrix}$             ≈⟨ ◦-cong (+-homo y z) refl ⟩
    ($\begin{pmatrix} \,\href{Algebra.Morphism.RingMonomorphism.html#3778}{\htmlId{3928}{\htmlClass{Bound}{\text{y}}}}\, \end{pmatrix}$ ⊕ $\begin{pmatrix} \,\href{Algebra.Morphism.RingMonomorphism.html#3780}{\htmlId{3936}{\htmlClass{Bound}{\text{z}}}}\, \end{pmatrix}$) ⊛ $\begin{pmatrix} \,\href{Algebra.Morphism.RingMonomorphism.html#3776}{\htmlId{3945}{\htmlClass{Bound}{\text{x}}}}\, \end{pmatrix}$       ≈⟨ distribˡ $\begin{pmatrix} \,\href{Algebra.Morphism.RingMonomorphism.html#3776}{\htmlId{3969}{\htmlClass{Bound}{\text{x}}}}\, \end{pmatrix}$ $\begin{pmatrix} \,\href{Algebra.Morphism.RingMonomorphism.html#3778}{\htmlId{3975}{\htmlClass{Bound}{\text{y}}}}\, \end{pmatrix}$ $\begin{pmatrix} \,\href{Algebra.Morphism.RingMonomorphism.html#3780}{\htmlId{3981}{\htmlClass{Bound}{\text{z}}}}\, \end{pmatrix}$ ⟩
    $\begin{pmatrix} \,\href{Algebra.Morphism.RingMonomorphism.html#3778}{\htmlId{3993}{\htmlClass{Bound}{\text{y}}}}\, \end{pmatrix}$ ⊛ $\begin{pmatrix} \,\href{Algebra.Morphism.RingMonomorphism.html#3776}{\htmlId{4001}{\htmlClass{Bound}{\text{x}}}}\, \end{pmatrix}$ ⊕ $\begin{pmatrix} \,\href{Algebra.Morphism.RingMonomorphism.html#3780}{\htmlId{4009}{\htmlClass{Bound}{\text{z}}}}\, \end{pmatrix}$ ⊛ $\begin{pmatrix} \,\href{Algebra.Morphism.RingMonomorphism.html#3776}{\htmlId{4017}{\htmlClass{Bound}{\text{x}}}}\, \end{pmatrix}$ ≈⟨ ∙-cong (*-homo y x) (*-homo z x) ⟨
    $\begin{pmatrix} \,\href{Algebra.Morphism.RingMonomorphism.html#3778}{\htmlId{4065}{\htmlClass{Bound}{\text{y}}}}\, \,\href{Algebra.Bundles.Raw.html#5431}{\htmlId{4067}{\htmlClass{Function Operator}{\text{*}}}}\, \,\href{Algebra.Morphism.RingMonomorphism.html#3776}{\htmlId{4069}{\htmlClass{Bound}{\text{x}}}}\, \end{pmatrix}$ ⊕ $\begin{pmatrix} \,\href{Algebra.Morphism.RingMonomorphism.html#3780}{\htmlId{4077}{\htmlClass{Bound}{\text{z}}}}\, \,\href{Algebra.Bundles.Raw.html#5431}{\htmlId{4079}{\htmlClass{Function Operator}{\text{*}}}}\, \,\href{Algebra.Morphism.RingMonomorphism.html#3776}{\htmlId{4081}{\htmlClass{Bound}{\text{x}}}}\, \end{pmatrix}$         ≈⟨ +-homo (y * x) (z * x) ⟨
    $\begin{pmatrix} \,\href{Algebra.Morphism.RingMonomorphism.html#3778}{\htmlId{4127}{\htmlClass{Bound}{\text{y}}}}\, \,\href{Algebra.Bundles.Raw.html#5431}{\htmlId{4129}{\htmlClass{Function Operator}{\text{*}}}}\, \,\href{Algebra.Morphism.RingMonomorphism.html#3776}{\htmlId{4131}{\htmlClass{Bound}{\text{x}}}}\, \,\href{Algebra.Bundles.Raw.html#5405}{\htmlId{4133}{\htmlClass{Function Operator}{\text{+}}}}\, \,\href{Algebra.Morphism.RingMonomorphism.html#3780}{\htmlId{4135}{\htmlClass{Bound}{\text{z}}}}\, \,\href{Algebra.Bundles.Raw.html#5431}{\htmlId{4137}{\htmlClass{Function Operator}{\text{*}}}}\, \,\href{Algebra.Morphism.RingMonomorphism.html#3776}{\htmlId{4139}{\htmlClass{Bound}{\text{x}}}}\, \end{pmatrix}$ ∎)

  distrib : _DistributesOver_ _≈₂_ _⊛_ _⊕_ → _DistributesOver_ _≈₁_ _*_ _+_
  distrib distrib = distribˡ (proj₁ distrib) , distribʳ (proj₂ distrib)

  zeroˡ : LeftZero _≈₂_ 0#₂ _⊛_ → LeftZero _≈₁_ 0# _*_
  zeroˡ zeroˡ x = injective (begin
    $\begin{pmatrix} \,\href{Algebra.Bundles.Raw.html#5483}{\htmlId{4392}{\htmlClass{Function}{\text{0#}}}}\, \,\href{Algebra.Bundles.Raw.html#5431}{\htmlId{4395}{\htmlClass{Function Operator}{\text{*}}}}\, \,\href{Algebra.Morphism.RingMonomorphism.html#4365}{\htmlId{4397}{\htmlClass{Bound}{\text{x}}}}\, \end{pmatrix}$     ≈⟨ *-homo 0# x ⟩
    $\begin{pmatrix} \,\href{Algebra.Bundles.Raw.html#5483}{\htmlId{4428}{\htmlClass{Function}{\text{0#}}}}\, \end{pmatrix}$ ⊛ $\begin{pmatrix} \,\href{Algebra.Morphism.RingMonomorphism.html#4365}{\htmlId{4437}{\htmlClass{Bound}{\text{x}}}}\, \end{pmatrix}$ ≈⟨ ◦-cong 0#-homo refl ⟩
    0#₂ ⊛ $\begin{pmatrix} \,\href{Algebra.Morphism.RingMonomorphism.html#4365}{\htmlId{4478}{\htmlClass{Bound}{\text{x}}}}\, \end{pmatrix}$    ≈⟨ zeroˡ $\begin{pmatrix} \,\href{Algebra.Morphism.RingMonomorphism.html#4365}{\htmlId{4496}{\htmlClass{Bound}{\text{x}}}}\, \end{pmatrix}$ ⟩
    0#₂            ≈⟨ 0#-homo ⟨
    $\begin{pmatrix} \,\href{Algebra.Bundles.Raw.html#5483}{\htmlId{4540}{\htmlClass{Function}{\text{0#}}}}\, \end{pmatrix}$         ∎)

  zeroʳ : RightZero _≈₂_ 0#₂ _⊛_ → RightZero _≈₁_ 0# _*_
  zeroʳ zeroʳ x = injective (begin
    $\begin{pmatrix} \,\href{Algebra.Morphism.RingMonomorphism.html#4628}{\htmlId{4655}{\htmlClass{Bound}{\text{x}}}}\, \,\href{Algebra.Bundles.Raw.html#5431}{\htmlId{4657}{\htmlClass{Function Operator}{\text{*}}}}\, \,\href{Algebra.Bundles.Raw.html#5483}{\htmlId{4659}{\htmlClass{Function}{\text{0#}}}}\, \end{pmatrix}$     ≈⟨ *-homo x 0# ⟩
    $\begin{pmatrix} \,\href{Algebra.Morphism.RingMonomorphism.html#4628}{\htmlId{4691}{\htmlClass{Bound}{\text{x}}}}\, \end{pmatrix}$ ⊛ $\begin{pmatrix} \,\href{Algebra.Bundles.Raw.html#5483}{\htmlId{4699}{\htmlClass{Function}{\text{0#}}}}\, \end{pmatrix}$ ≈⟨ ◦-cong refl 0#-homo ⟩
    $\begin{pmatrix} \,\href{Algebra.Morphism.RingMonomorphism.html#4628}{\htmlId{4735}{\htmlClass{Bound}{\text{x}}}}\, \end{pmatrix}$ ⊛ 0#₂    ≈⟨ zeroʳ $\begin{pmatrix} \,\href{Algebra.Morphism.RingMonomorphism.html#4628}{\htmlId{4759}{\htmlClass{Bound}{\text{x}}}}\, \end{pmatrix}$ ⟩
    0#₂            ≈⟨ 0#-homo ⟨
    $\begin{pmatrix} \,\href{Algebra.Bundles.Raw.html#5483}{\htmlId{4803}{\htmlClass{Function}{\text{0#}}}}\, \end{pmatrix}$         ∎)

  zero : Zero _≈₂_ 0#₂ _⊛_ → Zero _≈₁_ 0# _*_
  zero zero = zeroˡ (proj₁ zero) , zeroʳ (proj₂ zero)

  neg-distribˡ-* : (∀ x y → (⊝ (x ⊛ y)) ≈₂ ((⊝ x) ⊛ y)) → (∀ x y → (- (x * y)) ≈₁ ((- x) * y))
  neg-distribˡ-* neg-distribˡ-* x y = injective (begin
    $\begin{pmatrix} \,\href{Algebra.Bundles.Raw.html#5457}{\htmlId{5077}{\htmlClass{Function Operator}{\text{-}}}}\, \,\htmlId{5079}{\htmlClass{Symbol}{\text{(}}}\,\,\href{Algebra.Morphism.RingMonomorphism.html#5048}{\htmlId{5080}{\htmlClass{Bound}{\text{x}}}}\, \,\href{Algebra.Bundles.Raw.html#5431}{\htmlId{5082}{\htmlClass{Function Operator}{\text{*}}}}\, \,\href{Algebra.Morphism.RingMonomorphism.html#5050}{\htmlId{5084}{\htmlClass{Bound}{\text{y}}}}\,\,\htmlId{5085}{\htmlClass{Symbol}{\text{)}}}\, \end{pmatrix}$     ≈⟨ -‿homo (x * y) ⟩
    ⊝ $\begin{pmatrix} \,\href{Algebra.Morphism.RingMonomorphism.html#5048}{\htmlId{5121}{\htmlClass{Bound}{\text{x}}}}\, \,\href{Algebra.Bundles.Raw.html#5431}{\htmlId{5123}{\htmlClass{Function Operator}{\text{*}}}}\, \,\href{Algebra.Morphism.RingMonomorphism.html#5050}{\htmlId{5125}{\htmlClass{Bound}{\text{y}}}}\, \end{pmatrix}$       ≈⟨ ⁻¹-cong (*-homo x y) ⟩
    ⊝ ($\begin{pmatrix} \,\href{Algebra.Morphism.RingMonomorphism.html#5048}{\htmlId{5170}{\htmlClass{Bound}{\text{x}}}}\, \end{pmatrix}$ ⊛ $\begin{pmatrix} \,\href{Algebra.Morphism.RingMonomorphism.html#5050}{\htmlId{5178}{\htmlClass{Bound}{\text{y}}}}\, \end{pmatrix}$) ≈⟨ neg-distribˡ-* $\begin{pmatrix} \,\href{Algebra.Morphism.RingMonomorphism.html#5048}{\htmlId{5203}{\htmlClass{Bound}{\text{x}}}}\, \end{pmatrix}$ $\begin{pmatrix} \,\href{Algebra.Morphism.RingMonomorphism.html#5050}{\htmlId{5209}{\htmlClass{Bound}{\text{y}}}}\, \end{pmatrix}$ ⟩
    ⊝ $\begin{pmatrix} \,\href{Algebra.Morphism.RingMonomorphism.html#5048}{\htmlId{5223}{\htmlClass{Bound}{\text{x}}}}\, \end{pmatrix}$ ⊛ $\begin{pmatrix} \,\href{Algebra.Morphism.RingMonomorphism.html#5050}{\htmlId{5231}{\htmlClass{Bound}{\text{y}}}}\, \end{pmatrix}$   ≈⟨ ◦-cong (sym (-‿homo x)) refl ⟩
    $\begin{pmatrix} \,\href{Algebra.Bundles.Raw.html#5457}{\htmlId{5277}{\htmlClass{Function Operator}{\text{-}}}}\, \,\href{Algebra.Morphism.RingMonomorphism.html#5048}{\htmlId{5279}{\htmlClass{Bound}{\text{x}}}}\, \end{pmatrix}$ ⊛ $\begin{pmatrix} \,\href{Algebra.Morphism.RingMonomorphism.html#5050}{\htmlId{5287}{\htmlClass{Bound}{\text{y}}}}\, \end{pmatrix}$   ≈⟨ sym (*-homo (- x) y) ⟩
    $\begin{pmatrix} \,\href{Algebra.Bundles.Raw.html#5457}{\htmlId{5325}{\htmlClass{Function Operator}{\text{-}}}}\, \,\href{Algebra.Morphism.RingMonomorphism.html#5048}{\htmlId{5327}{\htmlClass{Bound}{\text{x}}}}\, \,\href{Algebra.Bundles.Raw.html#5431}{\htmlId{5329}{\htmlClass{Function Operator}{\text{*}}}}\, \,\href{Algebra.Morphism.RingMonomorphism.html#5050}{\htmlId{5331}{\htmlClass{Bound}{\text{y}}}}\, \end{pmatrix}$       ∎)

  neg-distribʳ-* : (∀ x y → (⊝ (x ⊛ y)) ≈₂ (x ⊛ (⊝ y))) → (∀ x y → (- (x * y)) ≈₁ (x * (- y)))
  neg-distribʳ-* neg-distribʳ-* x y = injective (begin
    $\begin{pmatrix} \,\href{Algebra.Bundles.Raw.html#5457}{\htmlId{5501}{\htmlClass{Function Operator}{\text{-}}}}\, \,\htmlId{5503}{\htmlClass{Symbol}{\text{(}}}\,\,\href{Algebra.Morphism.RingMonomorphism.html#5472}{\htmlId{5504}{\htmlClass{Bound}{\text{x}}}}\, \,\href{Algebra.Bundles.Raw.html#5431}{\htmlId{5506}{\htmlClass{Function Operator}{\text{*}}}}\, \,\href{Algebra.Morphism.RingMonomorphism.html#5474}{\htmlId{5508}{\htmlClass{Bound}{\text{y}}}}\,\,\htmlId{5509}{\htmlClass{Symbol}{\text{)}}}\, \end{pmatrix}$     ≈⟨ -‿homo (x * y) ⟩
    ⊝ $\begin{pmatrix} \,\href{Algebra.Morphism.RingMonomorphism.html#5472}{\htmlId{5545}{\htmlClass{Bound}{\text{x}}}}\, \,\href{Algebra.Bundles.Raw.html#5431}{\htmlId{5547}{\htmlClass{Function Operator}{\text{*}}}}\, \,\href{Algebra.Morphism.RingMonomorphism.html#5474}{\htmlId{5549}{\htmlClass{Bound}{\text{y}}}}\, \end{pmatrix}$       ≈⟨ ⁻¹-cong (*-homo x y) ⟩
    ⊝ ($\begin{pmatrix} \,\href{Algebra.Morphism.RingMonomorphism.html#5472}{\htmlId{5594}{\htmlClass{Bound}{\text{x}}}}\, \end{pmatrix}$ ⊛ $\begin{pmatrix} \,\href{Algebra.Morphism.RingMonomorphism.html#5474}{\htmlId{5602}{\htmlClass{Bound}{\text{y}}}}\, \end{pmatrix}$) ≈⟨ neg-distribʳ-* $\begin{pmatrix} \,\href{Algebra.Morphism.RingMonomorphism.html#5472}{\htmlId{5627}{\htmlClass{Bound}{\text{x}}}}\, \end{pmatrix}$ $\begin{pmatrix} \,\href{Algebra.Morphism.RingMonomorphism.html#5474}{\htmlId{5633}{\htmlClass{Bound}{\text{y}}}}\, \end{pmatrix}$ ⟩
    $\begin{pmatrix} \,\href{Algebra.Morphism.RingMonomorphism.html#5472}{\htmlId{5645}{\htmlClass{Bound}{\text{x}}}}\, \end{pmatrix}$ ⊛ ⊝ $\begin{pmatrix} \,\href{Algebra.Morphism.RingMonomorphism.html#5474}{\htmlId{5655}{\htmlClass{Bound}{\text{y}}}}\, \end{pmatrix}$   ≈⟨ ◦-cong refl (sym (-‿homo y)) ⟩
    $\begin{pmatrix} \,\href{Algebra.Morphism.RingMonomorphism.html#5472}{\htmlId{5701}{\htmlClass{Bound}{\text{x}}}}\, \end{pmatrix}$ ⊛ $\begin{pmatrix} \,\href{Algebra.Bundles.Raw.html#5457}{\htmlId{5709}{\htmlClass{Function Operator}{\text{-}}}}\, \,\href{Algebra.Morphism.RingMonomorphism.html#5474}{\htmlId{5711}{\htmlClass{Bound}{\text{y}}}}\, \end{pmatrix}$   ≈⟨ sym (*-homo x (- y)) ⟩
    $\begin{pmatrix} \,\href{Algebra.Morphism.RingMonomorphism.html#5472}{\htmlId{5749}{\htmlClass{Bound}{\text{x}}}}\, \,\href{Algebra.Bundles.Raw.html#5431}{\htmlId{5751}{\htmlClass{Function Operator}{\text{*}}}}\, \,\href{Algebra.Bundles.Raw.html#5457}{\htmlId{5753}{\htmlClass{Function Operator}{\text{-}}}}\, \,\href{Algebra.Morphism.RingMonomorphism.html#5474}{\htmlId{5755}{\htmlClass{Bound}{\text{y}}}}\, \end{pmatrix}$       ∎)

isRing : IsRing _≈₂_ _⊕_ _⊛_ ⊝_ 0#₂ 1#₂ → IsRing _≈₁_ _+_ _*_ -_ 0# 1#
isRing isRing = record
  { +-isAbelianGroup = isAbelianGroup R.+-isAbelianGroup
  ; *-cong           = *-cong R.*-isMagma
  ; *-assoc          = *-assoc R.*-isMagma R.*-assoc
  ; *-identity       = *-identity R.*-isMagma R.*-identity
  ; distrib          = distrib R.+-isGroup R.*-isMagma R.distrib
  } where module R = IsRing isRing

isCommutativeRing : IsCommutativeRing _≈₂_ _⊕_ _⊛_ ⊝_ 0#₂ 1#₂ →
                    IsCommutativeRing _≈₁_ _+_ _*_ -_ 0# 1#
isCommutativeRing isCommRing = record
  { isRing = isRing C.isRing
  ; *-comm = *-comm C.*-isMagma C.*-comm
  } where module C = IsCommutativeRing isCommRing